The production function is 𝑓(𝑥1, 𝑥2) = 𝑙𝑛(𝑥1 + 1) + 𝑙𝑛(𝑥2 + 1). Sketch the long-run supply curve. Are there corner solutions?
WLOG (without loss of generality) we can assume that 𝑤1 < 𝑤2. If the price is less than W1, for example what happens to x1*,x2*?
To sketch the long-run supply curve, we need to find the optimal inputs that maximize the production function subject to some input prices. Let's assume the input prices are 𝑤1 and 𝑤2 for 𝑥1 and 𝑥2 respectively.
To find the optimal inputs, we need to solve the following problem:
maximize 𝑓(𝑥1, 𝑥2) = 𝑙𝑛(𝑥1 + 1) + 𝑙𝑛(𝑥2 + 1)
subject to the constraint 𝑤1𝑥1 + 𝑤2𝑥2 ≤ 𝑃
where 𝑃 is the price.
We can solve this problem using calculus. Taking the partial derivatives of the production function with respect to 𝑥1 and 𝑥2, we have:
∂𝑓/∂𝑥1 = 1/(𝑥1 + 1)
∂𝑓/∂𝑥2 = 1/(𝑥2 + 1)
Setting these derivatives equal to zero and solving, we find that 𝑥1* = 𝑒^(𝑤2 - 𝑤1) - 1 and 𝑥2* = 𝑒^(𝑤1 - 𝑤2) - 1.
Now let's consider the case where the price 𝑃 is less than 𝑤1. In this case, the constraint 𝑤1𝑥1 + 𝑤2𝑥2 ≤ 𝑃 cannot be satisfied. Since we assume 𝑤1 < 𝑤2, the price is less than the cost of input 𝑥1.
As a result, there will be no positive levels of 𝑥1* and 𝑥2* that can maximize the production function. This means that there are no corner solutions in this case.
In summary, if the price is less than 𝑤1, the optimal levels of 𝑥1* and 𝑥2* will be zero due to input 𝑥1 being too expensive. There are no corner solutions in this scenario.
To sketch the long-run supply curve and determine if there are any corner solutions, we need to find the optimal values of x1 and x2 that maximize the production function f(x1, x2).
First, let's find the optimal values of x1* and x2* by maximizing the production function f(x1, x2). To do this, we can take the partial derivatives of f(x1, x2) with respect to x1 and x2, and set them equal to zero:
∂f/∂x1 = 1/(x1 + 1) = 0 (partial derivative with respect to x1)
∂f/∂x2 = 1/(x2 + 1) = 0 (partial derivative with respect to x2)
Solving these equations, we can see that x1* = x2* = -1. However, these values are not feasible as they are negative, and the production function f(x1, x2) is only defined for positive values of x1 and x2.
Since there are no feasible optimal values for x1 and x2 that maximize the production function, there are no corner solutions. This means that the long-run supply curve will be continuous and smooth.
To determine what happens to x1* and x2* when the price is less than W1, we need to consider the marginal productivities of x1 and x2, which can be found by taking the second partial derivatives of f(x1, x2):
∂^2f/∂x1^2 = -1/(x1 + 1)^2
∂^2f/∂x2^2 = -1/(x2 + 1)^2
Since the marginal productivities are negative for any positive values of x1 and x2, this implies that as the price decreases below W1, the optimal values of x1* and x2* will also decrease. In other words, when the price is less than W1, the firm will choose to produce less of both x1 and x2.
To sketch the long-run supply curve for the production function 𝑓(𝑥1, 𝑥2) = 𝑙𝑛(𝑥1 + 1) + 𝑙𝑛(𝑥2 + 1), we need to consider the optimal combination of inputs (𝑥1*, 𝑥2*) at different price levels.
To find the long-run supply curve, we first need to determine the optimal input levels for different price levels. This can be done by solving the profit maximization problem, which involves finding the values for 𝑥1* and 𝑥2* that maximize the profit function given a specific price level.
In this case, the profit function can be calculated as the difference between the revenue and the total cost. The revenue is the product of the quantity produced and the price, while the total cost is a function of the input levels and the input prices.
Since the specific input prices are not given in the question, we can assume that 𝑤1 represents the price of input 𝑥1, and 𝑤2 represents the price of input 𝑥2. Without loss of generality, we assume that 𝑤1 < 𝑤2.
Now, suppose the price is less than 𝑤1, implying that the price is insufficient to cover the cost of input 𝑥1. In this case, the optimal input level for 𝑥1, denoted as 𝑥1*, will be zero. This is because it would not make economic sense to use any of input 𝑥1 when the price is less than its cost.
However, since 𝑥1 is a component in the production function, setting 𝑥1* to zero will also affect the optimal input level for 𝑥2, denoted as 𝑥2*. The change in 𝑥2* will depend on the specific production function and the relationship between 𝑥1 and 𝑥2.
To determine the effect on 𝑥2*, we would need to take the partial derivative of the profit function with respect to 𝑥2 and evaluate it at 𝑥1* = 0. Then, we could analyze the sign of the derivative to understand whether 𝑥2* would increase or decrease when 𝑥1* is zero.
In summary, if the price is less than 𝑤1, the optimal input level for 𝑥1, 𝑥1*, would be zero. The effect on 𝑥2* would depend on the specific production function and the relationship between 𝑥1 and 𝑥2, which can be analyzed by taking the partial derivative of the profit function with respect to 𝑥2 and evaluating it at 𝑥1* = 0.